Problem: Perform the row operation, $R_3 \leftrightarrow R_1$, on the following matrix. $\left[\begin{array} {ccc} 3 & -2 & 1 & -4 \\ -3 & 2 & 10 & 0 \\ -6 & 4 & 7 & -3 \end{array} \right] $
Answer: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Switching the rows For the given matrix, $R_1$ and $R_3$ are given below. $R_1=\left[\begin{array} {ccc} 3 & -2 & 1 & -4 \end{array} \right] ~~~~~ R_3=\left[\begin{array} {ccc} -6 & 4 & 7 & -3 \end{array} \right]$ $\left[\begin{array} {ccc} 3 & -2 & 1 & -4 \\ -3 & 2 & 10 & 0 \\ -6 & 4 & 7 & -3 \end{array} \right] \xrightarrow{R_3 \leftrightarrow R_1}\left[\begin{array} {ccc} -6 & 4 & 7 & -3 \\ -3 & 2 & 10 & 0 \\ 3 & -2 & 1 & -4 \end{array} \right] $ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} -6 & 4 & 7 & -3 \\ -3 & 2 & 10 & 0 \\ 3 & -2 & 1 & -4 \end{array} \right] $